buffon s needle

8/11/2019 Buffon s Needle

1/20

The MathematicaJournal

Throwing Buffons

Needle withMathematica

Enis Siniksaran

It has long been known that Buffons needle experiments can be used to

estimate p. Three main factors influence these experiments: grid shape,grid density, and needle length. In statistical literature, several experi-ments depending on these factors have been designed to increase theefficiency of the estimators of p and to use all the information as fully aspossible. We wrote the package BuffonNeedle to carry out the most com-mon forms of Buffons needle experiments. In this article we review statisti-cal aspects of the experiments, introduce the package BuffonNeedle, discussthe crossing probabilities and asymptotic variances of the estimators, anddescribe how to calculate them using Mathematica.

Introduction

Buffons needle problem is one of the oldest problems in the theory of geometricprobability. It was first introduced and solved by Buffon [1] in 1777. As is wellknown, it involves dropping a needle of length lat random on a plane gridof parallel lines of width d>lunits apart and determining the probability of theneedle crossing one of the lines. The desired probability is directly related to thevalue of p, which can then be estimated by Monte Carlo experiments. This pointis one of the major aspects of its appeal. When pis treated as an unknown param-eter, Buffons needle experiments can be seen as valuable tools in applying the

concepts of statistical estimation theory, such as efficiency, completeness, andsufficiency. For instance, in order to obtain better estimators of p, Kendall andMoran [2] and Diaconis [3] examine several aspects of the problem with a longneedle (l>d). Morton [4] and Solomon [5] provide the general extension of theproblem. Perlman and Wishura [6] investigate a number of statistical estimationprocedures for pfor the single, double, and triple grids. In their study, they showthat moving from single to double to triple grid, the asymptotic variances of theestimators get smaller and hence more efficient estimators can be obtained.Wood and Robertson [7] introduce the concept of grid density and provide analternative idea. They show that Buffons original single grid is actually the most

efficient if the needle length is held constant (at the distance between lines on thesingle grid) and the grids are chosen to have equal grid density (i.e., equal length

The Mathematica Journal 11:1 2008 Wolfram Media, Inc.


2/20

of grid material per unit area). In [8], Wood and Robertson investigate the waysof maximizing the information in Buffons experiments.

We organize this article as follows. In the first three sections, we review Buffonsexperiments on single, double, and triple grids and their statistical issues. In thenext section, we introduce the features of the package BuffonNeedle. The

functions in the package implement Monte Carlo experiments for the three typesof grids. The results of each experiment are given in a table and in a picture.When the number of the needles thrown on each grid is large, very nice picturesexhibit the interface between chance and necessity. In the last two sections, wedescribe how to calculate the crossing probabilities in single- and double-gridexperiments and the asymptotic variances of the estimators for each grid using

Mathematica.

S ng e-Gr Exper ment

The single-grid form is Buffons well-known original experiment. A plane (tableor floor) has parallel lines on it at equal distances dfrom each other. A needleof length l(l


3/20

Let N1 be the number of times in n independent tosses that the needle crosses

any line. Then the proportion of crossings p`1, a point estimator of p1, becomes

p`1

=N1 n. Hence, we can write the point estimator of qin equation (2) as

(3)q`

=

p`1

2r=

N1

2 r n .

The random variable N1 is binomially distributed with the parameters n and p1.

q`

is the uniformly minimum variance unbiased estimator (UMVUE). Further-more, it is the maximum likelihood estimator (MLE) of qand hence has 100%

asymptotic efficiency in this experiment [6]. The variance of q`is then

(4)VarHq` L =Var N12r n

=p1H1 -p1L

4r2n=

qH1 - 2 rqL2 n r

=q

2n

1

r- 2q ,

which is minimized by taking ras close as possible to 1. Choosing needle length

l =d(r= 1) ensures this purpose. In this case, the variance of q`becomes

(5)VarHq` L = q2

2n

1

q- 2 .

In Buffons experiments, the parameter of main interest is p rather than q. The

estimator of this parameter, p`

= 1 q`, is called Buffons estimator and can beobtained from equation (3) as

(6)p =2r

p`1

.

It can also be expressed in terms of p`0

=N0 n

(7)p`

=2r

1 -p`0

,

where N0 is the number of times in n tosses that the needle does not cross any

line. Using equation (6) or (7) and Monte Carlo methods, we can obtain empir-

ical estimates of p for various values of r. The best estimate is expected at r= 1(l =d). Standard theory [11] ensures that Buffons estimator is an asymptoticallyunbiased, 100% efficient estimator of 1 q. Applying the delta method shows thatits asymptotic variance is

(8)AVarHp` L = p2

2nKp

r- 2O,

Throwing Buffons Needle With Mathematica 73



4/20

which is, as expected, minimized at r= 1. For this value of r, the asymptotic vari-ance of p

`is

(9)AVarHp` L = p2

2nHp - 2L.

If it is evaluated at p = 3.1416, we have

(10)AVarHp` L = 5.63n

.

Double-Grid Experiment

In the double-grid experiment, also called the Laplace extension of Buffonsproblem, a plane is covered with two sets of parallel lines where one set is orthog-onal to the other.

d

d

d d d

Figure 2. Buffons needles on a double grid.

In Figure 2, we see a double-grid plane and three needles of length lcrossingzero, one, and two lines. These crossing probabilities are

(11)

p 0 = 1 - rH4 - rLq,p 1 = 2rH2 - rLq,p 2 = r

2q.

Let Nibe the number of times in n tosses that the needle crosses exactly ilines(i= 0, 1, 2). Perlman and Wichura [6] showed that the random variable N1 +N2is distributed binomially with the parameters n and m qand is completely suffi-

cient for qwhere mis 4 r- r2. They also showed that the random variable

(12)q`

=

N1 +N2

m n

74 Enis Siniksaran



5/20

is the UMVUE and has 100% asymptotic efficiency with the variance

(13)VarHq` L = qn

l

m- q .

As in the case of a single grid, the variance of q

`

is minimized by r= 1, or equiv-alently by l =d. Replacing N1 +N2 with n -N0 in the right-hand side of equa-

tion (12), we have

(14)q`

=n -N0

m n=

1

m 1 -

N0

n=

1 -p`0

4r- r2.

Then Buffons estimator, p`

= 1 q` , can be expressed as

(15)p`

=4r- r2

1 -p

`

0

,

which can be used to obtain empirical estimates of p. By the delta method, wecan obtain the asymptotic variance of p

`as

(16)AVarHp` L = p2I4r- r2 - pM

n rHr- 4L ,which is minimized at r= 1. For this value of r, it becomes

(17)AVar

Hp`

L=

p 2

3 n

Hp - 3

L.

When evaluated at p = 3.1416, it is

(18)AVarHp` L = 0.466n

.

Compare the last equation with equation (10). Buffons estimator in the double-grid experiment is 5.63 0.466 > 12 times as efficient as that in the single-gridexperiment.

Triple-Grid Experiment

In the triple-grid experiment, a plane is covered with equilateral triangles of alti-

tude dand hence of side 2d 3 .




6/20

d

d

2d

3

Figure 3. Buffons needles on a triple grid.

Figure 3 shows a triple-grid plane and four needles of length lcrossing zero, one,two, and three lines. In [7], the crossing probabilities are given as

(19)

p 0 = 1 +r2

2-

3

2r 4 -

3

2r q,

p 1 = -5

4r2 +

3

2r 4 -

3

2r q,

p 2 = r2 -

3 3

4r2 q,

p 3 = -r2

4+

3 3

4r2 q.

Let Ni denote the number of times in n tosses that the needle crosses exactly ilines (i= 0, 1, 2, 3). For this experiment, Perlman and Wichura [6] investigated

the random variable N1 +N2 +N3 which is distributed binomially with the pa-

rameters n and a q - 1 2 where a = 3r2I4 - 3 r2M. They proposed, amongothers, the following unbiased estimator of qas a function of N1 +N2 +N3

(20)q`

=1

a

N1 +N2 +N3

n+

1

2.

76 Enis Siniksaran



7/20

By replacing N1 +N2 +N3 with n -N0, as in the other experiments, we obtainthe same estimator as a function ofN0

(21)q`

=1

a

n -N0

n+

1

2=

1

a

3

2-p

`0

.

The variance of q`is

(22)VarHq` L = H2a q - 3L H2 a q - 1L4 a2n

.

As in the cases of the single- and double-grid experiments, the variance of q`

isminimized by taking r= 1 (l =d).

From equation (21), Buffons estimator can be written as

(23)p`

=

2a

3 - 2p0

=

3rI8 - 3 rM2I3 - 2p0M .

For this experiment, the asymptotic variance of Buffons estimator is

(24)

AVarHp` L = -K2p2 K-3 3 + pO K-3 3 + 4 pOK-24 + 3 3 r+ 5 p rO K3 rK-8 + 3 rO + 2 p I2 + r2MOO K9 n rK54 K64 - 16 3 r+ 3 r2O + p2 K512 - 128 3 r-

270 r

2 +96 3 r

3 +27 r

4

O+

6p

K-

320 3+

168 r+ 234 3 r2 - 306 r3 + 27 3 r4OOO.For r= 1 and p = 3.1416, it is

(25)AVarHp` L = 0.015781n

.

Comparing this with equations (10) and (18), we can infer that Buffons estimatorin the triple-grid experiment is 0.466 0.015781 > 29 times as efficient as in thedouble-grid experiment and 5.63

0.015781 > 356 times as efficient as in the

single-grid experiment (see Table 1). Now, we can conclude that when weincrease the complexity of the grid, we can obtain tighter estimators of p. Woodand Robertson [7] investigated this conclusion. They introduced the notionof grid density, which is the average length of grid in a unit area and showed thatwhen the experiments are standardized, Buffons estimator in a single grid is themost efficient. In their approach, when d = 1, the single grid has unit grid den-sity, the double grid has grid density of two, and the triple grid has grid densityof three. Hence, the standardization of experiments corresponds to r= 1 in thesingle grid, r= 1 2 in the double grid and, finally, r= 1 3 in the triple grid.Replacing these values of rin equations (8), (16), and (24) and evaluating them at

p = 3.1416 yields the values of AVarHp`

L given in Table 2. As Wood and Rob-ertson claimed, the tightest estimator is obtained in the single-grid experiment.




8/20

Grid Type Single Hr= 1L Double Hr= 1L Triple Hr= 1LAVarHp` L 5.63 n 0.466 n 0.015781 n

Table 1. Asymptotic variances of Buffons estimator for three grids.

Grid Type Single Hr= 1L Double Hr= 1 2L Triple Hr= 1 3LAVarHp` L 5.63 n 7.85 n 5.91 n

Table 2. Asymptotic variances of Buffons estimator for three standardized grids.

The Package

The BuffonNeedle package is designed to throw needles on single, double, andtriple grids. Copy the file BuffonNeedle.m (see Additional Material) into theMathematica@AddOns @Applications folder. The following command loads theprogram.

In[18]:=


9/20

In[19]:= SingleGrid@10, .87D***Results of throwing 10 needles

of length l = 0.87.d on a single grid ***

ZeroCrossing

OneCrossing

Number of Crossings 4 6

Frequency Ratio 0.4000 0.6000

Theoretical Probability 0.4461 0.5539

Estimate of Pi:2.9

Out[19]=

In[20]:= DoubleGrid@15, .76D***Results of throwing 15 needles

of length l = 0.76.d on a double grid ***

Zero

Crossing

One

Crossing

Two

Crossings

Number of Crossings 2 11 2

Frequency Ratio 0.1333 0.7333 0.1333

Theoretical Probability 0.2162 0.6000 0.1839

Estimate of Pi:2.84123

Out[20]=




10/20

In[21]:= TripleGrid@12, .70D***Results of throwing 12 needles

of length l = 0.7.d on a triple grid ***

ZeroCrossing

OneCrossing

TwoCrossings

ThreeCrossings

Number of Crossings 2 8 2 0

Frequency Ratio 0.1667 0.6667 0.1667 0

Theoretical Probability 0.1107 0.5218 0.2874 0.08011


Out[21]=

In[22]:= SingleGrid@10 000, .91D***Results of throwing 10000 needles

of length l = 0.91.d on a single grid ***

Zero

Crossing

One

Crossing

Number of Crossings 4145 5855

Frequency Ratio 0.4145 0.5855

Theoretical Probability 0.4207 0.5793


Out[22]=

80 Enis Siniksaran



11/20

In[23]:= DoubleGrid@38 000, .85D***Results of throwing 38000 needles

of length l = 0.85.d on a double grid ***

ZeroCrossing

OneCrossing

TwoCrossings

Number of Crossings 5675 23 577 8748

Frequency Ratio 0.1493 0.6204 0.2302

Theoretical Probability 0.1477 0.6223 0.2300


Out[23]=

In[24]:= TripleGrid@28 000, 1D***Results of throwing 28000 needles

of length l =

1.d on a triple grid ***

Zero

Crossing

One

Crossing

Two

Crossings

Three

Crossings

Number of Crossings 97 7000 16 392 4508

Frequency Ratio 0.003464 0.2500 0.5854 0.1610

Theoretical Probability 0.003637 0.2464 0.5865 0.1635


Out[24]=




12/20

Calculating the Crossing Probabilities in Single andDouble Grids

In this section, we show how to calculate the crossing probabilities in single- anddouble-grid experiments usingMathematica.

Single-Grid Probabilities

In the single-grid experiment, two independent random variables with uniformdistribution are defined to determine the relative position of the needle to thelines: the distance Xof the needles midpoint to the closest line and the acuteangle aformed by the needle (or its extension) and the line (Figure 4). It is seenthat Xcan take any value between 0 and d 2 and acan take any value between 0and p 2. The density functions of Xand aare then given byIn[25]:= gdist1 =UniformDistributionB:0, d

2

>F;fX =PDF@gdist1, xD

Out[26]= 2d

0 x d

2

In[27]:= gdist2 =UniformDistributionB:0, Pi2

>F;fa =PDF@gdist2, aD

Out[28]= 2p

0 a p

2

Since Xand a are independent, the joint density function is the product of thedensity function of Xalone and the density function of aalone:

In[29]:= fX,a =2

d

2

p

Out[29]=

4

d p

for 0 x d2, 0 a p 2.From Figure 4, it is clear that the needle crosses the line when X l2sin a.The probability of this event is then

In[30]:= p1 = 0

p2

0

Hl2L*Sin@aD

fX,a xa

Out[30]=

2 l

d p

82 Enis Siniksaran



13/20

As there are two possible outcomes in the single-grid experiment, the probabilitythat the needle does not cross any line is given by

In[31]:= p0 =1 - p1

Out[31]= 1 -2 l

d p

which can alternatively be calculated by

In[32]:= p0 = 0

p2

Hl2L*Sin@aD

d2

fX,a xa

Out[32]= 1 -2 l

d p

The probabilities p0 and p1 can be written as a function of q and r as in

equation(1).

In[33]:= p0. Pi 1q

. l r * dOut[33]= 1 - 2 r q

In[34]:= p1. Pi 1q

. l r * dOut[34]= 2 r q

d

d

X

l2

l2

a

Figure 4. The random variables in the single-grid experiment.

Double-Grid Probabilities

In the double-grid experiment, three independent random variables with uni-form distribution can be defined to determine the relative position of the needleto the lines: the distance Xof the needles midpoint to the closest horizontalline, the distance Yof the needles midpoint to the closest vertical line, and theacute angle a formed by the needle and the horizontal line, as in Figure5. It isseen that Xand Y can take any value between 0 and d

2 and a can take any

value between 0 and p 2. The density functions of X, Y, and aare given by




14/20

In[38]:= gdist1 =UniformDistributionB:0, d2>F;

fX =PDF@gdist1, xDOut[39]= 2

d0 x

d

2

In[40]:= fY =PDF@gdist1, yDOut[40]= 2

d0 y

d

2

In[41]:= gdist2 =UniformDistributionB:0, Pi2

>F;fa =PDF@gdist2, aD

Out[42]= 2p

0 a p

2

As in the case of the single-grid experiment, the joint density function of X, Y,and ais the product of the density functions of X, Y, and a:

In[43]:= fX,Y,a =2

d

2

d

2

p

Out[43]=

8

d2 p

for 0 x d2, 0 y d2, 0 a p 2.In the double-grid experiment, there are four possible outcomes:

The needle crosses a horizontal line while not crossing a vertical line.

The needle crosses a vertical line while not crossing a horizontal line.

The needle crosses both a vertical line and a horizontal line or, equiva-lently, the needle crosses two lines.

The needle crosses neither a vertical line nor a horizontal line or,equivalently, the needle crosses no line.

The needle crosses a horizontal line but does not cross a vertical line whenX l2sin aand Y >l2cos a. The probability of this event is given byIn[44]:= p

XY =

0

p2

Hl2L*Cos@aD

d2

0

Hl2L*Sin@aD

fX,Y,a xya

Out[44]=

H2 d - lLld2 p

84 Enis Siniksaran



15/20

The needle crosses a vertical line but does not cross a horizontal line whenX >l2sin aand Y l 2cos a. The probability of this event is given byIn[45]:= p

X

Y=

0

p2

0

Hl2L*Cos@aD

Hl2L*Sin@aD

d2

fX,Y,axya

Out[45]= H2 d-

lLld2 p

Thus, the probability that the needle crosses exactly one line is

In[46]:= p1 =pXY + p

X

Y

Out[46]=

2H2 d - lLld2 p

The needle crosses both the vertical line and the horizontal line whenX l

2sin aand Y l

2cos a. The probability of this event is

In[47]:= p2 = 0

p2

0

Hl2L*Cos@aD

0

Hl2L*Sin@aD

fX,Y,a xya

Out[47]=

l2

d2 p

Finally, the needle crosses neither the vertical line nor the horizontal line whenX >l2sin aand Y >l2cos a. The probability of this event isIn[48]:= p0 =

0

p2

Hl2L*Cos@aD

d2

Hl2L*Sin@aD

d2

fX,Y,a xya

Out[48]= 1 +lH-4 d + lL

d2 p

The probabilities p0, p1, and p2can be written as functions of qand r, as in equa-

tion (11).

In[49]:= p0. Pi 1q

. l r * dSimplify

Out[49]= 1-

4 rq +

r2

q

In[50]:= p1. Pi 1q

. l r * dSimplifyOut[50]= -2H-2 + rLr q

In[51]:= p2. Pi 1q

. l r * dSimplify

Out[51]= r2

q




16/20

d

d

d d d

X

Ya

Figure 5. The random variables in the double-grid experiment.

Delta Method and Asymptotic Variance

Let a random variable Y be a function of the random variable X, that is,Y =HHXL. When the function HHXL is nonlinear, it may not be possible tocompute the true mean and the true variance of Y. One can, however, calculateestimates of the true mean and true variance. The delta method is a very usefulway to find such estimates [12, 13] and is based on the Taylor expansion about

the mean of X. Let the mean of Xbe m and the variance s2. Then the Taylor

expansion of the function HHXLabout mto the third term is(26)HHXL =HHmL +HHX- mL + 1

2H!HX - mL2.

Taking the expectation of both sides, we obtain the approximate mean of Y =HHXLas

(27)AMean@HHXLD =E@HHXLD =HHmL + 12

H!HmLs2.From the well-known identity of statistics

(28)Var@HHXLD =E8HHXL -E@HHXLD


17/20

in equation (29), substituting HHXL = p` , m = q, and s2 =VarHq` L fromequations(5), (13), and (22), we can obtain the asymptotic variances of p

`in equa-

tions (8), (16), and (24) for each grid.

Alternatively, asymptotic variances of p`can be computed as follows [7, 11]

(30)AVar@p` D = @HHqLD2

n IHqL .Here IHqLis the Fisher information number, given by

(31)IHqL = i

Api!HqLE2

piHqL ,

where piHqL is the probability that the needle crosses ilines. These probabilitiesgiven in equations (1), (11), and (19) actually define a list for each grid as follows:

In[53]:= ProbSingle =81 - 2 r q, 2 r q;

From equations (30) and (31), one can define the following functions to obtainthe asymptotic variances:

In[56]:= deriv@expr_, var_D:= HD@expr, varDL2

HexprL ;fisherInfo@list_, var_D:=

Total@Table@deriv@list@@iDD, varD,8i, Length@listD


18/20

In[61]:= aVar@ProbTriple, qD

Out[61]= 1 n q 4 27 r4

16Jr2 - 34

3 r2 qN +

27 r4

16J- r24

+ 3

43 r2 qN

+

9 r2 K4 - 3 r2

O2

4K1 + r22

- 3

2rK4 - 3 r

2O qO

+

9 r2 K4 - 3 r2

O2

4K- 5 r24

+ 3

2rK4 - 3 r

2O qO

Substituting q = 1 pand factoring the expressions, we haveIn[62]:= aVar@ProbSingle, qD . q 1

pFactor

aVar@ProbDouble, qD . q 1pFactor

aVar@ProbTriple, qD . q 1p

Factor

Out[62]=

p2 Hp - 2 rL

2 n r

Out[63]= -

p2 Hp - 4 r + r2L

nH-4 + rLrOut[64]= -J2 p2 J-3 3 + pN J-3 3 + 4 pN

J-24 + 3 3 r + 5 p rN J4 p - 24 r + 3 3 r2 + 2 pr2NN J9 n rJ3456 - 1920 3 p + 512 p 2 - 864 3 r + 1008 pr -

128 3 p2 r + 162 r2 + 1404 3 pr2 - 270 p2 r2 -

1836 pr3 + 96 3 p2 r3 + 162 3 pr4 + 27 p 2 r4NNwhich were previously given in equations (8), (16), and (24), respectively. Forr= 1 and p = 3.1416, we obtain the same values given in Table 1 for each grid.

In[65]:= aVar@ProbSingle, qD . q 1p

. r 1. p 3.1416

Out[65]=

5.6336

n

In[66]:= aVar@ProbDouble, qD . q 1p

. r 1. p 3.1416

Out[66]=

0.465848

n

88 Enis Siniksaran



19/20

In[67]:= aVar@ProbTriple, qD . q 1p

. r 1. p 3.1416

Out[67]=

0.0157809

n

For the standardized experiments of Wood and Robertson [7], we can obtain theasymptotic variances given in Table 2 by substituting r= 1, 1 2, and 1 3 forsingle, double, and triple grids, respectively.

In[68]:= aVar@ProbSingle, qD . q 1p

. r 1. p 3.1416

Out[68]=

5.6336

n

In[69]:= aVar@ProbDouble, qD . q 1p

. r 12

. p 3.1416

Out[69]=

7.84835

n

In[70]:= aVar@ProbTriple, qD . q 1p

. r 13

. p 3.1416

Out[70]=

5.90518

n

Acknowledgments

I would like to thank the reviewers whose comments led to a substantial improve-ment in this article. I would also like to thank my colleague, M. H. Satman, forcreating some of the functions in the package BuffonNeedle. Thanks to Dr. Aylin

Aktukun for reading and commenting on numerous versions of this manuscript.This work is supported by the Scientific Research Projects Unit of IstanbulUniversity.

References

1 G. L. Buffon, Essai darithmtique morale, Histoire naturelle, gnrale, et particulire,Supplment 4, 1777 pp. 685|713.

2 M. G. Kendall and P. A. P. Moran, Geometrical Probability, New York: Hafner, 1963.

3 P. Diaconis, Buffons Problem with a Long Needle, Journal of Applied Probability, 13,1976 pp. 614|618.

4 R. A. Morton, The Expected Number and Angle of Intersections Between RandomCurves in a Plane,Journal of Applied Probability, 3, 1966 pp. 559|562.

5 H. Solomon, Geometric Probability, Philadelphia: Society for Industrial and Applied

Mathematics, 1978.




20/20

6 M. D. Perlman and M. J. Wichura, Sharpening Buffons Needle,American Statistician,29(4), 1975 pp. 157|163.

7 G. R. Wood and J. M. Robertson, Buffon Got It Straight, Statistics and ProbabilityLetters, 37,1998 pp. 415|421.

8 G. R. Wood and J. M. Robertson, Information in Buffon Experiments, Journalof Statistical Planning and Inferences, 66,1998 pp. 21|37.

9 M. R. Spiegel, Probability and Statistics, Schaums Outline of Probability and Statistics,New York: McGraw-Hill, 1975 pp. 67|68.

10 J. V. Uspensky, Introduction to Mathematical Probability, New York: McGraw-Hill, 1937,p. 252.

[11] C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed., New York: Wiley &Sons, 1973.

12 G. W. Oe ert, A Note on t e De ta Met o ,Amer can Stat st c an, 46, 1992 pp. 27|29.

13 J. A. Rice, Mathematical Statistics and Data Analysis, 2nd ed., Belmont, CA: DuxburyPress, 1994 p. 149.

Additional Material

BuffonNeedle.m

Available at www.mathematica-journal.com/issue/v11i1/download.

About the Author

Enis Siniksaran is an assistant professor at Istanbul University, Department of Economet-rics. His research interests include geometric approaches to statistical methods, robuststatistical methods, and the methodology of econometrics. He has been working withMathematica since 1998. For details on his recent work dealing with the geometry ofclassical test statistics, in which all symbolic and numerical computations and graphicalwork are done using Mathematica, see library.wolfram.com/infocenter/Articles/5962.

Enis SiniksaranIstanbul Universitesi, Iktisat Fak., Ekonometri BolumuBeyazt, 34452, Istanbul, Turkey

[email protected]

90 Enis Siniksaran

Th M th ti J l 11 1 2008 W lf M di I

buffon s needle

Documents